"Пить таблетки" (literally "drink pills") is a commonly used phrase. But why is it "пить" ("drink")? Naturally, pills are solid and thus can't be "drunk" in a traditional sense. There is phrase "есть таблетки" ("eat pills"), but it's more of a slang.
My best guess is that the phrase came from the time when most if not all medicine was in liquid form: sirups, infusions, etc. - but I'd like somebody with a better knowledge to confirm/deny it.